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Ceramic honeycomb structures have been used successfully as catalyst supports in gasoline-powered vehicles for the past ten years. They are currently the leading candidate for trapping and oxidizing the carbonaceous particulate emissions in diesel-powered vehicles. In both of these applications the long term durability of the ceramic substrate is of prime importance. This, in turn, depends on the physical properties of cellular structure, cyclic nature of service loads and design of the mounting assembly. This paper examines the nature and dependence of both the mechanical and thermal stresses in the substrate on its geometry, properties, mounting parameters, and the operating conditions. It also compares the observed failure modes with those predicted by the theory. The paper concludes with a set of recommendations for optimal systems design and acceptable operating conditions which will promote the long term durability of the ceramic substrate.